Formation of primordial black holes from density perturbations in the early universe

Question

Recently I've been reading Carr and Hawking paper on the formation of primordial black holes "Primordial black holes in the universe" , but I couldn't understand part of it. It says that:

A region that is about to collapse has also an upper limit to its size at the moment at which it begins to contract. To see how this arises, consider a spacelike hypersurface orthogonal to the matter flow which crosses the region at the moment when the rate of expansion is zero. The $R^{00}—\frac{1}{2}g^{00}R=8\pi T^{00}$ constraint equation implies that the 3-geometry of this hypersurface has positive curvature of order $\mu$ in the region where the rate of expansion is zero. If this positive curvature extended over a sufficiently large region, the spacelike hypersurface would close up on itself to form a disconnected compact 3-space of radius about $\mu^{-1/2}$. In this case the region would form a separate closed universe which was completely disconnected from our Universe. Such a situation would not correspond to a black hole formed by collapse of matter in our Universe.

why a close separate sub-universe cannot end up into a black hole? isn't it like that by close sub-universe we mean that this is a region where is under collapse and is separate from the hubble flow?

and I also read in the Sasaki review article the use of the expression: "separate universe approximation breaking down" for this situation, what does it mean?

and also it will be nice if someone give an explanation on what does "re-entering hubble horizon or re-collapsing" exactly means. (the stress in on the 're-...' ! "

at the end a brief view of PBHs formation will be helpful.

Answer 1

The Schwarzschild solution for a black hole is a vacuum static solution that does not apply in the regions filled with matter. The Friedmann solution is a solution for uniform and homogenous matter that may be expanding, contracting, or momentarily between the expansion and contraction.

If the universe expands in such a way that the density is uniformly higher in a sufficiently large region, this region can be approximately described by the Friedmann solution for that region. If the average density there is sufficient and the size is large enough, the region may become closed.

In contrast, a black hole does not represent a closed region, but only a causally asymmetric region. Per the no-hair theorem, we can observe a black hole by its mass, momentum, spin, and charge. A black hole is attractive, things can fall toward it. However, a closed region is closed completely, there are no geodesics leading to it, because such a region, unlike a black hole, would not have an "edge" or "border" or "trapped surface" like the event horizon of a black hole. This region would be completely closed, "looped onto itself", with no connection, interaction, or relation to our universe. Because there is no trapped surface, the no-hair theorem would not apply to this region. It would be unobservable and for all practical purposes non-existent in our view of the world, a completely different universe with its own spacetime disconnected from ours.

What may be causing a confusion in understanding the quoted text is the reference to "hypersurface" that may superficially sound like "horizon". There "hypersurface" actually refers to the entire volume of the region or more precisely, to a timeslice of the region.

The quoted text refers to a region just starting to collapse. However, conceptually, a higher density region does not necessarily have to be collapsing, but may become closed while still expanding. Although, once it is closed, it is causally disconnected from us, so there is no meaning in viewing his region as "expanding" or "contracting" at our moment of "now".

Re-entering the Hubble horizon refers to the areas previously expanding faster than light that are slowing down and getting back inside caught up by the Hubble sphere that keeps expanding at the speed of light.

Re-collapsing refers to the expansion slowing down and turning to a collapse.

A "a brief view of PBHs formation" would be better addressed in a separate question.